Transcript Document 7699272

LESSON 11: INVENTORY MODELS
(DETERMINISTIC)
Outline
•
•
•
•
•
•
•
Hierarchy of Production Decisions
Inventory Control Questions
Inventory Control Costs
The Economic Order Quantity (EOQ) model
The EOQ Model Costs
The EOQ
Some Important Characteristics of the EOQ Cost
Function
1
Hierarchy of Production Decisions
• In Chapter 2, Lessons 4-7, we discuss demand
forecasting
• Forecasting provides the demand of products in the
near future
• In Chapter 3, Lessons 8-10, we discuss aggregate
planning.
• Aggregate production plan is translated into a detail
product-wise production plan by Master Production
Schedule (MPS).
• The next question in the hierarchy of decision making
is inventory control.
2
Hierarchy of Production Decisions
Forecast of Demand
Aggregate Planning
Master Production Schedule
Inventory Control
Operations Scheduling
Vehicle Routing
3
Inventory Control Questions
• Inventory control questions:
– Raw materials, components (subassemblies), and
finished goods may be purchased or produced inhouse.
– Timing and amount of purchase and production must
be carefully planned. There are two major questions:
1. How much (lot size)?
How much to purchase or produce?
2. When?
When to purchase or produce?
4
Inventory Control Questions
• There can be two extreme strategies:
– Small lot sizes and too frequent inventory
replenishment
• This strategy yields higher ordering costs (if the
items are purchased) or setup costs (if the
items are produced).
• The demand forecast is for a short term and it
is more accurate.
– Large lot sizes and too infrequent inventory
replenishment
5
Inventory Control Questions
– Large lot sizes and too infrequent inventory
replenishment
• This strategy requires a large investment in the
inventory and yields higher inventory holding
costs.
• The demand forecast is for a long term and it is
less accurate.
6
Inventory Control Costs
• Inventory holding costs, h  Ic
– Opportunity cost of capital
– Cost of storage space
– Taxes and insurance against fire, theft, and other
losses
– Breakage, spoilage, deterioration and obsolescence
– Example of calculation of inventory holding costs
Cost of capital 15%
Taxes and insurance 2%
Storage 5%
Breakage/spoilage 3%
Total 25%
7
Inventory Control Costs
• Inventory holding costs, h  Ic
– Notation:
I = Annual interest rate
c = Dollar value of one unit of inventory
h = holding cost in terms of dollars per unit per year
– Then, we have the relationship
h  Ic
8
Inventory Control Costs
• Order costs or setup costs, K
– If the items are purchased, a fixed ordering cost may be
incurred each time an order is placed. Similarly, if the
items are produced, a fixed setup cost may be incurred
each time the production facility is set up to produce the
item.
– Some examples are bookkeeping expense, order
processing fees, transportation costs, receiving costs,
handling costs, etc.
9
Inventory Control Costs
• Order costs or setup costs, K
– If there is a variable part of the order cost/setup that
depends on the number of units ordered/produced, the
variable cost is usually not considered in the order cost.
Instead, the variable cost may be included in the cost of
the item.
– For example, the salary paid to the purchasing clerk
does not depend on the number of times orders are
placed. Such a cost is an overhead expense, and not a
part of the ordering cost.
– Notation:
K = ordering/setup cost per order/setup.
10
Inventory Control Costs
• Penalty costs, p
– Shortages occur when the demand exceeds the amount
of inventory on hand. One of two types of costs is
charged depending on whether a shortage results in
loss of sales or not:
• Backorder - if the excess demand is backlogged and
fulfilled in a future period, a backorder cost is charged
(bookkeeping and/or delay costs).
• Lost sales - if the excess demand is lost because the
customer goes elsewhere, the lost sales is charged.
The lost sales include goodwill and loss of profit
margin. So, penalty cost = selling price - unit variable
cost + goodwill, if there exists any.
11
Inventory Control Costs
• Penalty costs, p
– In Chapter 4, Lessons 11-15, we assume that the
demands are known and fixed and shortages will not
take place. So, penalty costs are not considered.
– In Chapter 5, Lessons 16-20, we assume that the
demands are uncertain and shortages may occur. So,
penalty costs are considered in Chapter 5.
12
The EOQ Model
• Major assumptions
1. Demand is known and fixed (uniform). The rate of
demand is  units per year. This assumption is relaxed
in Chapter 5, Lessons 16-20, Stochastic Inventory
Models.
2. The cost parameters, unit cost, c holding cost, h and
ordering cost K are known and fixed. Given that the
inventory control decisions are made for a short term,
it’s likely that the costs will not change during the
planning period. Still, this assumption is a simplification.
The costs parameters may change over time.
3. Shortages are not permitted. This assumption is relaxed
in Chapter 5, Lessons 16-20, Stochastic Inventory
Models.
13
The EOQ Model
• Major assumptions
4. The inventory level increases instantaneously at one
point of time when an order is received. This assumption
is appropriate in the context of purchasing. The EPQ
model, discussed in Lesson 12, considers a gradual
increase in the inventory level. The EPQ model is
appropriate in the context of production.
5. There is no price discount for large order sizes. This
assumption is relaxed later. See Lesson 13, the EOQ
with price break.
14
The EOQ Model
• Constant order size and inventory cycle
– Since there is no uncertainty, it’s optimal to plan an order
receipt only when the inventory level reaches zero.
– Suppose that there is no inventory in the beginning. So,
an order will be received in the beginning.
– Since the demand and cost parameters do not change
over time
• If it’s optimal to order Q units in the beginning, it’s
also optimal to order Q units next time.
– The above observation provides two important concepts.
One is the constant order size and the other is the
inventory cycle.
15
The EOQ Model
• Constant order size and inventory cycle
– The order size, Q is chosen to minimize the total
inventory control costs. The formula for optimal order
quantity is given later.
– At the start of each inventory cycle, the inventory level
is Q . The inventory level decreases uniformly. In the
end of the inventory cycle the inventory level is zero. So,
the length of the cycle is the length of time over which
the demand is Q . Thus, the length of the cycle in years
is
Q
T

16
The EOQ Model
Demand
rate
Inventory Level
Order qty, Q
Reorder point, R
0
Lead
time
Order
Order
Placed Received
Lead
Time
time
Order
Order
Placed
Received
17
The EOQ Model
• Lead time and reorder point
– Sometimes, there may be a lead time associated with
the orders. The lead time,  is the length of time
between order placement and order receipt.
– The presence of lead time requires the order be placed
some time before it is needed. For example, if there is a
lead time of 2 days, the order must be placed when the
inventory is sufficient to meet the demand for 2 days.
– Reorder point,
is the inventory level at the time of
order placement.
– Since, reorder point must cover the lead time demand,
R  
R
18
The EOQ Model Costs
Inventory level uniformly varies from 0 to Q
Q
So, the average inventory 
2
Q
So, the annual inventory holding cost  h
2

The annual number of orders 
Q

So, the annual ordering cost  K
Q
The annual cost of buying the items, c is
independen t on Q and is not considered .
hQ K
Hence, the total annual cost, TC 

2
Q
19
The EOQ Model Costs
Slope = 0
Annual
cost ($)
Minimum
total cost
Total Cost
hQ K
TC 

2
Q
hQ
Holding Cost =
2
K
Ordering Cost =
Q
Optimal solution, Q*, Economic Order Quantity (EOQ)
Order Quantity, Q
20
The EOQ
As the annual demand  and cost parameters c, h and K
are fixed (see major assumptions), it’s important to analyze
the effect of order quantity on the annual ordering, holding
and total costs. The previous slide shows such an effect.
K
Annual ordering cost,
decreases as the order quantity
Q
increases
hQ
Annual holding cost,
increases as the order quantity
2
increases
hQ K
The total cost, TC 
. The total cost curve is nearly

2
Q
U-shaped.
The total cost is minimum for some order quantity that’s
neither too small nor too large.
21
The EOQ
hQ K
Question : Find Q that minimizes TC 

2
Q
d TC 
h K
2 K
Answer : Set
 0 or,  2  0 or, Q 
d Q 
2 Q
h
This order quantity is called Economic Order quantity (EOQ)
and denoted by Q * .
2 K
Hence, EOQ, Q 
minimizes TC.
h
*
22
The EOQ
• Recall that there are two major inventory control questions,
how much and when. The EOQ model answers these
questions as follows.
• How much to order?
– Order
2 K
EOQ, Q 
h
*
• When to order?
– When the inventory on hand reaches the reorder point
R  
23
Example 1: R & B beverage company has a soft drink
product that has a constant annual demand rate of 3600
cases. A case of the soft drink costs R & B $3. Ordering
costs are $20 per order and holding costs are 25% of the
value of the inventory. R & B has 250 working days per
year, and the lead time is 5 days. Identify the following
aspects of the inventory policy:
a. Economic order quantity
24
b. Reorder point
c. Cycle time
d. Total annual cost
25
• Some notes on Example 1:
– The total cost curve is flat near the EOQ value. So, the
total cost does not change much because of a little change
in the order quantity from the EOQ value. See the
discussion under some important characteristics. Since the
number of cases of soft drinks is a whole number, the EOQ
value has been rounded to the nearest integer.
– It’s not a coincidence that the annual holding cost is nearly
the same as the annual ordering cost. If the EOQ units are
ordered the annual ordering cost is the same as the
annual holding cost. See the total cost curve and the
discussion under EOQ model costs and some important
characteristics. The little difference between two costs is
due to the rounding.
26
Some Important Characteristics of the EOQ
Cost Function
• At EOQ, the annual holding cost is the same as annual
ordering cost.
hQ* h 2 K
Kh
Annual holding cost 


2
2
h
2
K
Annual ordering cost  * 
Q
K

2 K
h
Kh
2
hQ* K
Total annual cost 
 *  2 Kh
2
Q
27
Some Important Characteristics of the EOQ
Cost Function
• If the order quantity is near the EOQ value, the total cost
does not change from the optimal value.
– See the discussion under the EOQ model costs and
the figure showing various costs against order quantity.
The total cost curve is flat near EOQ. This supports the
point.
– The EOQ policy usually provides a good decision even
when the cost parameters are little off than the
assumed values (and, therefore, the EOQ value is
incorrect). So, the EOQ model is insensitive to errors.
See text pp. 208-209 for a detail discussion. In this
note, the insensitivity is shown with an example. 28
Some Important Characteristics of the EOQ
Cost Function

3600
3400
3600
3600
I
25%
25%
35%
25%
K
20
20
20
30
Q*
438
426
370
537
Opt
Cost
329
319
389
402
Cost
with
Q =438
329
320
394
411
29
Some Important Characteristics of the EOQ
Cost Function
• If the order quantity is near the EOQ value, the total cost
does not change from the optimal value.
– Consider the Example 1 data. The assumed values of
the annual demand, holding cost and ordering costs
are 3600 units, 25% and $20/order respectively. So,
the EOQ=438 units.
30
Some Important Characteristics of the EOQ
Cost Function
• If the order quantity is near the EOQ value, the total cost
does not change from the optimal value.
– Suppose that the correct value of the annual demand is
3400 units. So, the correct EOQ=426 units and the
optimal total annual cost is $319. If the decision maker,
being unaware of the correct value of the annual
demand, uses an order quantity of 438 units, the total
annual cost will be $320 which is less than 0.30% off
the optimal value of $319.
– Similarly, the order quantity of 438 units, does not
produce a large error when holding cost changes to
31
35% or ordering cost to $30/order.
READING AND EXERCISES
Lesson 11
Reading:
Section 4.1 - 4.5 , pp. 194-208 (4th Ed.), pp. 183-200
(5th Ed.)
Exercise:
10 and 12, p. 210 (4th Ed.), pp. 201-202 (5th Ed.)
32